This post duplicates an example in Richard M. Goodwin's Chaotic Economic Dynamics (Oxford University Press, 1990). At least, I think it does, but without the typographic errors that I think are in Goodwin's book. My Figure 3 is Goodwin's Figure 2.1, and my Figure 2 is Goodwin's Figures 2.2, and 2.3.
I have no plans to prepare a Part 2 to post later. But I describe in the conclusion below why there should be a Part 2.
2.0 Supply and Demand
This model is a partial equilibrium model with well-behaved supply and demand curves. It is an internal exploration of a mainstream textbook model. The demand curve shows the price that must instantaneously prevail if the quantity on the market is to be sold:
Time is discrete in this model, and the supply curve contains a lag. Firms plan the quantity to supply in the next period based on the price in this period:
It's easy enough to solve for equilibrium, in which the quantity supplied and the quantity demanded are equal and do not change through time. Equation 3 gives the equilibrium quantity:
|Figure 1: Supply and Demand|
3.0 Numerical Exploration of a Special Case
Goodwin considers the special case where the parameters b and e are both zero. Under this special case, the difference equation becomes considerably simplified:logistic equation. As a start at exploring the dynamics of Equation 7, consider the case where a is unity, c is 3/5, and d is 21/20. (Update: Parameters were originally specified incorrectly.) Figure 2 shows time paths for two arbitrary initial values of the redefined quantity. Both time paths converge to a two-period limit cycle. The upper extreme of the blue time path continually falls, while the upper extreme of the red path continually rises. They meet in the limit at one point in the limit cycle. The lower extremes, shown in FIgure 2, converge to the other point in the limit cycle.
|Figure 2: Time Paths of Redefined Quantity|
|Figure 3: Phase Space for These Paths|
|Figure 4: Phase Space for Period Four Cycle|
|Figure 5: Phase Space Showing Chaos|
The above exposition begins with a common introductory model in economists. And it ends with mathematical chaos. Chaos is shown in a special case in which both the demand and the supply curves go through the origin. A supply curve going through the origin, although a special case, is quite reasonable in economics. It is not economically sensible for the demand curve to go through the origin.
If there were to be a part 2 for this post, it would demonstrate the possibility of chaos in an economically relevant parameter range. One would want the demand curve to be declining throughout the first quadrant as well as intersecting the price axis at a strictly positive price. And one would want the strange attractor to lie entirely in the first quadrant for the price and untransformed quantity variables. But I haven't done enough numerical exploration yet.